Derivative of eᶜᵒˢˣ⋅cos(eˣ)

let's now use what we know about the chain rule in the product rule to take the derivative of an even weirder expression so we're going to take the derivative we're going to take the derivative of e to the cosine of x times the cosine of e to the X so let's take the derivative of this so we can view this as the product of two functions so the product rule tells us that this is going to be the derivative with respect to X of e to the cosine of X e to the cosine of x times cosine times cosine of e to the X plus plus the first function just e to the cosine of X e to the cosine of x times the derivative of the second function times the derivative with respect to X of cosine of e to the X cosine of e to the X and so we just need to figure out what these two derivatives are and so you can imagine the chain rule might be applicable here so let me make it clear this we got from the product rule product product rule but then to evaluate each of these derivatives we need to use the chain rule so let's think about this a little bit so the derivative let me copy and paste this so I don't have to rewrite it so copy and paste so let's think about what the derivative of e to the cosine of X is e to the cosine of X so we could view our outer function is e to the something as e to the something and the derivative of e to the something with respect to something is just going to be e to that something so it's going to be e to the cosine of X so let me do that in that same blue color so it's going to be e actually let me do that actually let me do it in a new color let me do it in magenta so the derivative of e to the something with respect to something is just e to the something it's just e to the cosine of X and we have to multiply that times the derivative of the something with respect to X so what's the derivative of cosine of X with respect to X well that's just negative sine of X so it's times negative sine of X and so we figured out this first derivative let me make it clear this right over here is the derivative of e to the cosine of X derivative of e to the cosine of X with respect to with respect to cosine of X and this right over here this right over here is the derivative of cosine of X with respect with respect to X and we just took the product of the two that's what the chain rule tells us fair enough now let's figure out this derivative out here so we want to find the derivative with respect to X of cosine eetu the X so then once again let me copy and paste it so we need to figure out this thing right over here so first just like we did we're just going to apply the chain rule again we need to figure out the derivative of cosine of something in this case e to the X with respect to that something so this is going to be equal to derivative of cosine of something with respect to that something is equal to the negative sine of that something negative sine of EDA of e to the X once again we can view this as the derivative of cosine of e to the X with respect to with respect to e to the X and then we multiply that times the derivative of the something with respect to X so let me do this in this I'm running out of colors let me do this in this green color so times the derivative of e to the X with respect to X is just e to the X so that right over there is the derivative of e to the X with respect with respect to X and so we're essentially done we just have to substitute what we found using the chain rule back into our original expression the derivative of this business up here is going to be equal to let me just copy and paste everything just to make everything nice and nice and clean so copy and paste so that is going to be equal to is going to be equal to this times cosine X so this is going to be let's see we could put the e to the X out front we could put the negative out front so we could write it as negative e to the cosine X e so the cosine x times sine of x times sine of x times cosine of e to the x times cosine of e to the X so that's this first term here plus plus e to the cosine x times all of this stuff and so let's see we could put the negative out front again so let's put that negative out front so we have a negative negative e to the cosine of x times e to the X so I can write it this way e to the x times e to the cosine X and you could simplify that or combine it since you're multiplying two things with the same base but I'll just leave it like this like this e to the x times e to the cosine x times the sine we already have the negative so then we have sine of e to the X sine of e to the X so let me write it over here times sine sine of e to the X we had negative sine e to the x times e to the X negative sine of e to the x times e to the X and then that was multiplied by e the cosine of X so we have the exact same thing right over here and we're done